Fundamental theorem of calculus — Safekipedia Discoverer

The fundamental theorem of calculus is a theorem that connects two important ideas in math: differentiating a function and integrating a function. Differentiating helps us understand how a function changes at each point, like finding the slope of a line. Integrating helps us find the total area under a curve or the build-up of small effects over time. This theorem tells us that these two processes are opposites of each other, working like a pair of scissors.

The first part of the theorem says that for a continuous function f, we can find something called an antiderivative by integrating the function over an interval that changes. This means we can use integration to build back the original function from its rate of change.

The second part of the theorem tells us that to find the total area under a function f over a fixed interval, we just need to look at how an antiderivative F changes between the two ends of that interval. This makes calculating areas much easier if we can find the antiderivative using symbolic integration, instead of relying on more difficult numerical integration methods.

History

Sketch of geometric proof

The first fundamental theorem of calculus shows how two important ideas in math are connected: finding slopes and finding areas under curves.

Imagine you have a curve on a graph. You can create an "area function" that tells you the area under the curve up to any point. By looking at small strips of area and comparing them to rectangles, we find that the slope of this area function at any point is exactly the value of the original curve at that point. This means that taking the derivative of an area (an integral) gives you back the original function. In other words, derivatives and integrals are like opposite operations, undoing each other. This is the core idea of the Fundamental Theorem of calculus.

Intuitive understanding

The fundamental theorem of calculus tells us that integration and differentiation are opposite operations. Think of it like this: if you know how fast a car is moving (its speed) at every moment, you can figure out how far it has traveled by adding up all the tiny distances it moves each second. This adding up of tiny distances is what we call integration.

The theorem also says that if you know the total distance traveled up to any point, the rate at which that distance is changing at that point is exactly the speed of the car. So, integrating gives us the total distance, and differentiating tells us the speed.

Formal statements

The Fundamental Theorem of Calculus has two main parts that connect two big ideas in math: derivatives and integrals.

Fundamental theorem of calculus (animation)

The first part says that if you have a special kind of function, you can find its antiderivative by integrating it. This means that integrating and then taking the derivative brings you back to your original function.

The second part shows that if you know an antiderivative of a function, you can easily find the value of a definite integral by using the antiderivative at the endpoints of the interval. This part is useful even if the function isn’t perfectly continuous everywhere.

antiderivative definite integrals

Proof of the first part

To understand the first part of the Fundamental Theorem of Calculus, we look at a special function ( F ) defined by an integral. For any two points ( x_1 ) and ( x_1 + \Delta x ), the difference between ( F ) at these points can be expressed using the integral of the function ( f ) between them.

By the mean value theorem for integration, there is a point ( c ) between ( x_1 ) and ( x_1 + \Delta x ) where this integral equals ( f(c) ) times ( \Delta x ). As ( \Delta x ) gets very small, the ratio of the change in ( F ) to ( \Delta x ) approaches ( f(c) ). Because ( c ) stays close to ( x_1 ) as ( \Delta x ) shrinks, this ratio ultimately equals ( f(x_1) ), showing that the derivative of ( F ) at ( x_1 ) is ( f(x_1) ). This connects integration and differentiation, showing they are inverse operations.

mean value theorem for integration squeeze theorem

Proof of the corollary

The proof shows that if we have a function f that doesn’t stop or change suddenly between two points, a and b, we can find another function F that helps us calculate the total area under f between those points. This is done by comparing two ways to describe the same idea: one using a special function G that adds up small pieces of f, and the other using F, which we already know works well. By looking closely, we see they are almost the same, just shifted by a fixed amount. This leads to a very useful result: the total area under f from a to b is simply the difference between the values of F at those two points.

Main article: mean value theorem
Main article: constant function

Proof of the second part

A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function.

This proof uses a method called Riemann sums. It connects two big ideas in math: taking the slope of a line (called differentiation) and finding the area under a curve (called integration).

The proof starts with a special math rule called the mean value theorem. This rule helps us break down a complicated problem into smaller, easier pieces. By adding up these small pieces and making them closer and closer together, we get a very accurate answer. In the end, this shows that the difference between two values of a function is the same as the integral — the total area under the curve — between those two points. This links differentiation and integration together in a clear way.

F ( b ) − F ( a ) = ∑ i = 1 n [ F ( x i ) − F ( x i − 1 ) ] . {\displaystyle F(b)-F(a)=\sum _{i=1}^{n}[F(x_{i})-F(x_{i-1})].}

F ( b ) − F ( a ) = ∑ i = 1 n [ f ( c i ) ( Δ x i ) ] . {\displaystyle F(b)-F(a)=\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].}

Relationship between the parts

The two parts of the Fundamental Theorem of Calculus are closely connected. If we know one part, we can often understand the other better. For example, if we have a function f and we know one of its antiderivatives G, then we can describe the integral of f from a point a to another point x using G. This shows that integrating and finding antiderivatives are almost opposite operations.

However, it’s important to remember that not every function has a simple antiderivative we can write down. Some functions can be integrated even though we can’t find a simple expression for their antiderivatives. Also, some functions might have antiderivatives but still not be integrable in the usual sense. This shows how the two parts of the theorem fit together in a careful way.

Examples

The Fundamental Theorem of Calculus shows how two main ideas in math—finding slopes of lines and finding areas under curves—are connected. It tells us that these two processes are opposites of each other.

One example shows how to find the area under a curve between two points. By using a special formula, we can find the exact area without complicated calculations. Another example shows how the theorem helps us understand how functions change over time, linking the idea of accumulation to the idea of instantaneous change.

Variations in terminology

Different math books use different names for the parts of the fundamental theorem of calculus. Some call the first part the "first fundamental theorem," which says that if you have a continuous function, you can find a new function by integrating it over an interval.

Other books might call this the "fundamental theorem" and have a different name for the second part. Even old math history books change how they name these ideas, so it can get a little confusing! But no matter what it's called, the ideas connect slopes and areas in smart ways.

Generalizations

The Fundamental Theorem of Calculus can be extended beyond continuous functions. Even if a function isn’t continuous everywhere, we can still find important results. For example, if a function is continuous at just one point within an interval, we can create a new function by integrating it, and this new function will be differentiable at that point.

The theorem also works for more complex functions and integrals. In higher dimensions, ideas from the theorem help us understand how averages of functions behave, and they connect to important results like the divergence theorem and gradient theorem. These extensions show how deeply the Fundamental Theorem of Calculus influences many areas of mathematics.

Main article: Lebesgue's differentiation theorem

Main articles: Divergence theorem, Gradient theorem

Further information: Generalized Stokes theorem